PHYS 1444 Section 003 Lecture 18 Thursday Nov

PHYS 1444 – Section 003 Lecture #18 Thursday, Nov. 3, 2011 Dr. Jaehoon Yu • • • Torque on a Current Loop Magnetic Dipole Moment Magnetic Dipole Potential Energy Sources of Magnetic Field Due to Straight Wire • Forces Between Two Parallel Thursday, Nov. Wires 3, PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 1

Announcements • Quiz #3 – Beginning of the class coming Tuesday, Nov. 8 – Covers: CH 26. 5 through what we finish Today (CH 28. 4? )! • Reading Assignments – CH 27. 6 – 27. 9 • Bring your special project at the end of the class! Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 2

Torque on a Current Loop • What do you think will happen to a closed rectangular loop of wire with electric current as shown in the figure? – It willmagnetic rotate! Why? The field exerts a force on both vertical sections of wire. – Where is this principle used in? • Ammeters, motors, volt-meters, speedometers, etc • The two forces on the different sections of the wire exerts net torque to the same direction about the rotational axis along the symmetry axis of the wire. • What happens when the wire turns 90 degrees? – It will not turn unless. PHYS the 1444 -003, direction of the current changes Fall 2011 3 Thursday, Nov. 3, 2011 Dr. Jaehoon Yu

Torque on a Current Loop • So what would be the magnitude of this torque? – What is the magnitude of the force on the section of the wire with length • Fa=Ia. B a? • The moment arm of the coil is b/2 – So the total torque is the sum of the torques by each of the forces • Where A=ab is the area of the coil loop – What is the total net torque if the coil consists of N loops of wire? 1444 -003, Fallthe 2011 field – If the coil makes an. PHYS angle θ w/ Dr. Jaehoon Yu Thursday, Nov. 3, 2011 4

Magnetic Dipole Moment • The formula derived in the previous page for a rectangular coil is valid for any shape of the coil • The quantity NIA is called the magnetic dipole moment – It is considered a vectorof the coil • Its direction is the same as that of the area vector A and is perpendicular to the plane of the coil consistent with the right-hand rule – Your thumb points to the direction of the magnetic moment when your finer cups around the loop in the direction of the wire – Using the definition of magnetic moment, the torque can be written in vector form Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 5

• Magnetic Dipole Potential Energy Where else did you see the same form of the torque? – Remember the torque due to electric field on an electric dipole? – The potential energy of the electric dipole is – • How about the potential energy of a magnetic dipole? – The work done by the torque is – – If we chose U=0 at θ =π/2, then C=0 Thursday, Nov. 3, PHYS 1444 -003, Fall 2011 – Thus the potential energy is 2011 Dr. Jaehoon Yu 6

Example 27 – 12 Magnetic moment of a hydrogen atom. Determine the magnetic dipole moment of the electron orbiting the proton of a hydrogen atom, assuming (in the Bohr model) it is in its ground state with a circular orbit of radius 0. 529 x 10 -10 m. What provides the centripetal The Coulomb force? So we can obtain the speed of the electron from Solving for v Since the electric current is the charge that passes through the given point per unit time, we can obtain theofcurrent Since the area the orbit is A=πr 2, we obtain the hydrogen magnetic moment Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 7

The Hall Effect • What do you think will happen to the electrons flowing through a conductor immersed in a magnetic field? – Magnetic force will push the electrons toward one side of the conductor. Then what happens? • – A potential difference will be created due to continued accumulation of electrons on one side. Till when? Forever? – Nope. Till the electric force inside the conductor is • This is called the Hall Effect equal and opposite to the magnetic force – The potential difference produced is called • The Hall emf – The electric field due to the separation of charge Hall field, Thursday, Nov. 3, is called the PHYS 1444 -003, Fall. E 2011 H, 2011 Dr. Jaehoon Yu and it points to the direction opposite 8

The Hall Effect • In equilibrium, the force due to Hall field is balanced by the magnetic force evd. B, so we obtain • and • The Hall emf is then – Where l is the width of the conductor • What do we use the Hall effect for? – The current of negative charge moving to right is equivalent to the positive charge moving to the left – The Hall effect can distinguish these since the direction of the Hall field or direction of the Hall emf is opposite – Since the magnitude of the Hall emf is proportional to the magnetic field strength can measure the B-field strength Thursday, Nov. 3, • Hall probe 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 9

Sources of Magnetic Field • We have learned so far about the effects of magnetic field on electric currents and moving charge • We will now learn about the dynamics of magnetism – How do we determine magnetic field strengths in certain situations? – How do two wires with electric current interact? – What is the general approach to finding the connection between current and magnetic field? Thursday, Nov. 3, PHYS 1444 -003, Fall 2011 10 2011 Dr. Jaehoon Yu

Magnetic Field due to a Straight Wire • The magnetic field due to the current flowing through a straight wire forms a circular pattern around the wire – What do you imagine the strength of the field is as a function of the distance from the wire? • It must be weaker as the distance increases – How about as a function of current? • Directly proportional to the current – Indeed, the above are experimentally verified • This is valid as long as r << the length of the wire – The proportionality constant is μ 0/2π, thus the field strength becomes PHYS 1444 -003, 2011 – μ is the permeability of free. Fall space Thursday, Nov. 3, 0 2011 Dr. Jaehoon Yu 11

Example 28 – 1 Calculation of B near wire. A vertical electric wire in the wall of a building carries a dc current of 25 A upward. What is the magnetic field at a point 10 cm dueformula north for of this wire? field near a Using the magnetic straight wire So we can obtain the magnetic field at 10 cm away as Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 12

• Force Between Two Parallel Wires We have learned that a wire carrying the electric current produces magnetic field • Now what do you think will happen if we place two current carrying wires next to each other? – They will exert force onto each other. Repel or attract? – Depending on the direction of the currents • This was first pointed out by Ampére. • Let’s consider two long parallel conductors separated by a distance d, carrying currents I 1 and I 2. • At the location of the seconductor, the magnitude of the magnetic field produced by I 1 is Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 13

• Force Between Two Parallel Wires The force F by a magnetic field B on a 1 wire of length l, carrying the current I 2 when the field and the current are perpendicular to each other is: – So the force per unit length is – This force is only due to the magnetic field generated by the wire carrying the current I 1 • There is the force exerted on the wire carrying the current I 1 by the wire carrying current I 2 of the same magnitude but in opposite direction • So the force per unit length is • Thursday, about the direction of the force? Nov. 3, PHYS 1444 -003, Fall 2011 If. How the currents are in the same direction, the attractive force. 2011 Dr. Jaehoon Yu 14 If opposite, re

Example 28 – 5 Suspending a wire with current. A horizontal wire carries a current I 1=80 A DC. A second parallel wire 20 cm below it must carry how much current I 2 so that it doesn’t fall due to the gravity? The lower has a mass of 0. 12 g per meter length. is the gravitational Down to the center of the Whichofdirection Earth force? This force must be balanced by the magnetic force exerted on the wire by the first wire. Solving for I 2 Thursday, Nov. 3, 2011 PHYS 1444 -003, Fall 2011 Dr. Jaehoon Yu 15

Operational Definition of Ampere and Coulomb • The permeability of free space is defined to be exactly • The unit of current, ampere, is defined using the definition of the force between two wires each carrying 1 A of current and separated by 1 m – So 1 A is defined as: the current flowing each of two long parallel conductors 1 m apart, which results in a force of exactly 2 x 10 -7 N/m. • Coulomb is then defined as exactly 1 C=1 A s. Thursday, Nov. 3, this way since PHYS 1444 -003, Fall 2011 • We do it the electric current is 2011 Dr. Jaehoon Yu 16